## Ergodicity and parallel universes

13 July, 2014 at 16:37 | Posted in Statistics & Econometrics | 3 Comments

Consider the following experiment … Say you have an unbiased die and you ask your friend Sam to throw the die a thousand times. Now you compute the average of the thousand rolls (by calculating the probability of each face showing up, multiplying by the weight of each face, summing all the values and dividing by 1000). This value should converge to 3.5 by the law of large numbers. From a temporal perspective, this is one universe with Sam rolling the die sequentially. This is called the time average since each subsequent roll is sequential.

Now lets say you have the good fortune of having a thousand friends who exist in real life (and not on Facebook) who would be so kind to help with the next piece of the experiment. You give them each a similar unbiased die, and ask each of them to roll the die just once, for a cumulative 1000 rolls. You compute the average of the 1000 rolls and find that this average too converges to 3.5. This is the ensemble average.

This system is said to be ergodic because the time average is equal to the ensemble average. Easy right? Not so fast.

See the problem is that most economists tend to view the world as being ergodic, when the real world is far from it. Ergodic systems are by definition zero sum. Everything moves around but over the long run usually cancels out. There can thus be no growth in an ergodic system.

Let’s look at the problem by considering how to treat the expected value of wealth in the real world. To do so let us do the following thought experiment …

I give you an unbiased die and ask you to roll it. If the die comes up with a 6, I will give you 10 times your total net worth. If you roll anything other than a 6, you give me all your assets (your house, car, investments etc etc – just your assets, not your liabilities). Do you take the bet? If you’ve taken a statistics or finance class, you know that the expected value of the bet is 0.83 (-1W*5/6 + 10W*1/6, where W is your initial wealth). Logic would dictate that you take the bet since the expected value is positive. But most people are hesitant when it comes to this bet in particular. Why is that? …

The problem lies with how we think about the averages. When we computed the ensemble average above, we easily assumed that we live in parallel universes – 5 universes where we go broke and 1 where we come out on top. We then mixed these universes together to get our ensemble average of 0.83. And therein lies the problem – we don’t live in a world with parallel universes. We live in a world which is non-ergodic – where wealth can grow and you can easily go broke.

In order to look at the problem the right way, we need to compute the time average of the returns. To do so, you don’t have to assume that you live in 6 parallel universes but rather just one, where you roll the die 6 times, one after the other. To compute the time average returns, you take the six outcomes (the order doesn’t matter) and take the 6th root to give you the answer. The time average in this case is negative. Thus, the ensemble average in the first case tends to hide the real fact that you could easily lose all your wealth since the probability of loss is greater than the probability of coming out on top.

The difference between the time average and the ensemble average is generally small for systems where the volatility is small. But, the effect becomes more and more pronounced when volatility increases. So the next time you think about computing expected values, think about whether you need to compute the time average or the ensemble average. Also,consider whether the system you are dealing with is ergodic or non-ergodic.

Intellectual Entropy

Paul Samuelson once famously claimed that the “ergodic hypothesis” is essential for advancing economics from the realm of history to the realm of science. But is it really tenable to assume – as Samuelson and most other neoclassical economists – that ergodicity is essential to economics? The answer can only be – as the article above shows and as I have argued here, here, here, here and here – NO WAY!

1. I don’t want to be too quixotic
but if the world is egrotic
then like janus faces
I can be in two places
thats enough to make time neurotic (he, he)

• Lovely 🙂

2. Professor, it’s all a strawman argument. Expectations converge to ensemble averages only at the limit of large numbers. Laymen know that by common sense but some professors do not know it although it is in the Probability books some place (ex. Papoulis, Prob and Stat, p. 138).

“To compute the time average returns, you take the six outcomes (the order doesn’t matter) and take the 6th root to give you the answer.The time average in this case is negative. ”

What is this? Any game with a positive and a negative outcome will have a negative time average. Does this mean then nobody should ever take a bet when there is a possible loss in a binary outcome game? Let’s go back to the caves then.